Semiintegral Electroanalysis: Studies on the Neopolarographic Plateau Masashi Goto and Keith B. Oldham Trent University, Peterborough, Ontario, Canada
Simple theory predicts that neopoiarograms should have horizontal plateaus, but experimental curves usually display an upward slope. Theory is developed which seeks to explain departures from horizontality in terms of electrode curvature and double-layer charging. Four potential-time programs are treated: simple ramp, capped ramp, step, and pulse; the exact or approximate response of the semiintegrated current is predicted for each of these four cases. Experiments on the reduction of TI(I) and Fe(lll), and with supporting electrolyte alone, lend credence to the theory. Techniques for correcting experimental results for sphericity and nonfaradaic effects are discussed.
The simple theory of semiintegral electroanalysis predicts neopolarograms (and m u s . t curves) with flat horizontal plateaus ( I ) , but in practice such plateaus are seldom exactly horizontal. Here, we investigate the two effects which are believed responsible for this departure from horizontality. Though the theory is derived on the basis of reversible electrochemical behavior, and our experimental examples fall into the class of reversible electrode reactions, most of the conclusions of this article apply more generally. Wherever a restriction t o reversible processes applies, this has been stated explicitly. (Definitions of symbols used throughout are given in the Appendix.)
THEORY Throughout this article, we shall be concerned with voltammetric processes which occur a t a hanging mercury drop electrode in contact with a quiet solution containing a concentration C of a n electroreducible species Ox plus a n excess of inert supporting electrolyte. Initially ( a t time t = 0) the electrode is a t a potential Eo positive enough t h a t the electroreduction Ox(so1n)
+ ne-
+
Rd(so1n or amal)
(1)
does not occur. Subsequently, the potential E is programmed in one of the four ways depicted in Figure 1 so that, for part a t least of the time t > 0, virtually complete concentration polarization of Ox exists at the electrode. We shall be concerned with the ways in which the current i, its semiintegral 1 m E -d-’”’ dt-112
E = E , - Et,
E
= constant
(2)
it is predicted ( 2 )t h a t E and m are interrelated by
E
=
E,,,
+ nF
In(-)
(3)
an equation which predicts t h a t m acquires the asymptotic value m, as E becomes sufficiently negative. This equation may be rephrased as
or as
m. + t a n h ( g [ t
m = 2 (1
- t,,J)}
(5)
Equations 3 through 5 apply to reversible electroreductions only. However, the sigmoid shapes portrayed in the second and third diagrams of Figure 1 apply equally to irreversible and reversible processes. Theory and experiment (3) show that, as in classical polarography, reversible and irreversible neopolarographic waves are qualitatively similar in shape, the irreversible waves being somewhat less steep. For the capped ramp,
E
=
E , - Et
+ &[t - t f ] H ( t f )
(6)
simple theory shows t h a t Equations 3 through 5 still apply The f i us. t response to t h e step signal
E
= E,
- [E,, - E,]H(O)
(7:
m
(8)
is itself the step =
m,H(O)
L e . , the semiintegral jumps immediately to the constant
m, value where it remains indefinitely. One may regard a step as a capped-ramp signal in which the ramp rate is infinite; in fact, Equations 7 and 8 are respectively the & m limits of Equations 6 and 5 . The signal-independence of the m us. E relationship still applies but the “neopolarogram” is degenerate in t h e step case (see the ninth diagram in Figure 1) since the potential moves instantaneously from Eo to Ef without lingering a t any intermediate value. The pulse signal
-
and its integral
(the charge), vary during these processes. Figure 1 also shows the m responses predicted by simple theory for the four different voltage signals. For the ramp signal, (1) M GrennessandK B Oldham,Anal C h e m , 44, 1211 (1972)
1522
again evokes a n m response, m = m,H(O) - m,H(t,), which mirrors t h e signal. According to simple theory, the semiintegral should return exactly and permanently to zero after the pulse. Because -1.00 Itanh ( x ) I -0.98 for x 5 -2.3, it follows from Equation 4 t h a t if Ell2 - E I4.6RT/nF, then (2) M Goto and K B Oldham, Anal Chem , 45, 2043 (1973) (3) M Goto and K B Oldham, recent work submitted for publication in Anal Chem
ANALYTICAL CHEMISTRY, VOL. 46, NO. 11, SEPTEMBER 1974
I f
I2
I 2 0
/ 0
0
1
I
I
I
2
4
6
8
1 - 0
io
1 isec)
Semiintegrated current vs. time curves to concentrations in 0 . l M K N O s in potential step Figure 2.
(C)
TI-
at lower
Concentration ( m M ) , electrode area ( c m 2 ) , and step potential (mV vs. SCE): ( I ) 0.500. 0.0158, -200 lo -600, ( 1 1 ) 1.00. 0.115, ca. 0 to -600
(d)
Four ways in which the potential may b e programmed in semiintegral electroanalysis together with the predictions (according to the simplest theory) of t h e resulting variation of t h e semiintegral m as a function of time and of potential Figure 1.
0 (a) I S a simple ramp in which the potential € 0 at t 5 0 is made progressively ever more negative the resulting neopolarograms show a sigmoid variation from rn = 0 to rn = a constant m, With the capped ramp ( b ) . the potential remains constant at a rather negative final value Er for times t 2 t f , the neopolarograms closely resemble those obtalned with the simple ramp In the step ( c ) the potential is suddenly changed at f = 0 from the initial E o value to the final Ef value the semiintegral executes a similar step to the rn, value The pulse ( d ) differs from the step in that the potential returns to its initial value after a pulse duration t,, the value of the semiintegral is predicted for reversible processes to return to zero for t > t,, In all diagrams the shaded area represents the potential region in which Ox is virtually irreducible while the crosshatched area corresponds to potentials at which concentration polarization of Ox I S virtually complete
0.99 rn, 5 rn 5 m,. T h a t is (for T = 298 K ) the neopolar-
ographic plateau should (for reversible processes only) be essentially flat once a potential about (120/n) millivolts more negative t h a n El,* has been passed. For irreversible processes ( 3 ) ,a somewhat greater potential span is needed before the neopolarographic plateau is reached but, once reached, it is predicted to be just as flat. Experimental neopolarograms, however, generally have a sloping plateau, m continuing to increase with time (or potential), as is evident in t h e neopolarograms reproduced in references I and 2 Less frequently (see Figure 2 of this article) t h e plateau may exhibit a minimum. We believe t h a t these effects arise from two factors which are ignored in the simple theory: the presence of nonfaradaic current and the curvature of the electrode. The first of these effects is always present in semiintegral electroanalysis, though it may be trivial if electroactive concentrations are large. The second effect is always present, its seriousness depending on t h e magnitude of the L parameter, where r is the electrode radius. Fortunately i t is more or less possible to investigate each effect independently of the other and we shall discuss each effect separately before briefly considering their combined influence on neopolarography.
m/r
w I .
I
I
0
Figure 3. T h e
I
1
I 1 1, o r is
I
i
l
l
theoretical m n f vs. t curves for the four voltage
signals ( a ) : ramp: ( b ) : capped ramp: ( c ) : step: and ( d ) :pulse. The curves have been drawn to scale for the following approximate values: c = 20 fiF/ cm2, A = 3.0 X crn’, E o - Et = 0.50 V , E = 0.10 V/sec, t f = t, = 5.0 sec and with axis intervals of 1 second and 25 nanoamplornbs
Nonfaradaic Contribution. In addition to the passage of charge by virtue of t h e reduction process, reaction 1, charge will inevitably cross t h e electrode in consequence of the charging of the interfacial double-layer, whenever the electrode potential is changed. The magnitude of this (cathodic) nonfaradaic charge is easily calculated from the equation q n f = cA[Eo - E ] , if the double layer capacitance cA can be assumed constant. Since m is the semiderivative of q , the nonfaradaic contribution to m is given by
m,,f = c
d”’
. i p { ~ ~
E}
(10)
and is readily calculable for each of t h e four potential signals. For the simple ramp, combination of Equations 2 and 10 gives
and predicts a nonfaradaic component which increases continuously, as shown by curve ( a ) in Figure 3.
A N A L Y T I C A L C H E M I S T R Y , VOL. 46,
NO. 11,
SEPTEMBER 1974
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“T
I \
I
,
f
i ;
I 0
I ‘D
Figure 4. The shapes predicted for rn vs. t curves with step (c) and pulse ( d ) signals if nonfaradaic contribution were the only effect complicating simple behavior The “cusps” are characteristic of this effect. Following t = 0 and (in the pulse case) t = tl,, the m vs. f curve is actually infinitely steep; the dashed line represents the response of an instrument (such as a penand-ink recorder) which cannot follow such rapid excursions
To solve the corresponding capped-ramp problem, one must make use of the rule ( 4 )
for the differintegration of the product of an arbitrary function and the Heaviside function. Using this rule with Equations 6 and 10, we find
for the capped-ramp case. In this instance, the nonfaradaic contribution to m commences to decrease once tf is passed, as shown in curve ( b ) of Figure 3. The step signal case is easily treated. From Equations 7 and 10,
showing t h a t the nonfaradaic contribution, theoretically initially infinite, continuously declines. See curve (c) of Figure 3 for a graph of this decline, which may alternatively be derived from a careful consideration of the E m limit of Equation 12. Likewise for the pulse we find, by combining Equations 10 and 9 and invoking rule 11, that
-
amplomb, is obtainable from Figure 3 for the values quoted in its legend. With the additional typical value D = 1.1 x 10-5 cm2 sec-I, we find from the formula defining m, that m,/nC = A F d F = 10 amplomb cm3 equivalent-1. Hence, the faradaic semiintegral is predicted t o exceed the nonfaradaic m for all electroactive concentrations in excess of about 10-8 equivalent cm-3 (lO-5M if n = 1). With solution in the millimolar concentration range, therefore, the nonfaradaic contribution should not normally interfere with semiintegral electroanalysis. However, as curves ( c ) and ( d ) of Figure 3 demonstrate, the nonfaradaic component is extremely large for a short time after the imposition of a step or a pulse and after the termination of the latter. Since the experimentally measured response in semiintegral electroanalysis is the sum, m = n, + G f ,of the faradaic and nonfaradaic components, we would expect m cs t plateaus to have the shapes shown in Figure 4, for a step or a pulse signal, were nonfaradaic contribution the only effect causing non-horizontality of the plateau. Though there are exceptions, the double layer capacitance a t a mercury/solution interface is not usually markedly affected by the presence of small concentrations of electroactive solutes. Therefore, a simple method of correcting for nonfaradaic contribution is to follow the standard polarographic practice of subtracting a “blank” polarogram--1 e , a curve obtained under conditions identical apart from the absence of Ox. The subtraction of a blank has the added advantage of correcting also for m contributions due to the presence of traces of foreign electroreducible substances in the base electrolyte. We summarize the findings of this section as follows: 1) Nonfaradaic contribution to m values is expected to be a serious complication only a t sub-millimolar concentration levels. 2 ) With capped-ramp or step signals, the nonfaradaic contribution can, in principle, be made arbitrarily small by waiting long enough. Of course, one can in practice only wait as long as the onset of convection permits. 3) With step or pulse signals, the plateaus are predicted to display a n initial “cusp,” as in Figure 4. 4 ) Subtraction of a blank will, in almost all cases, completely correct for nonfaradaic effects. These four findings apply to reversible and irreversible processes alike. Electrode Curvature. The simple theory of semiintegral electroanalysis is based on the equation pair
where COand CO’ are the surface concentrations of Ox and Rd. These relationships are derived for, and are strictly only applicable to, a planar electrode of area A . For a spherical electrode with radius r, Equations 13 and 14 are to be replaced by ( 5 )
where g is a geometric parameter given by 1 if Rd is soluble in solution -1 if Rd is soluble in mercury
.={
as depicted in curve ( d ) of Figure 3. I n this case the behavior is initially similar to the step but for t > t,, the nonfaradaic semiintegral is continuously negative, approaching zero asymptotically a t large t values. It is of interest to compare values of m, and m,,,. A typical order of magnitude for the latter quantity, h fc 10-7
Of the four signals (ramp, capped-ramp, step, and pulse), it is the third which is simplest to handle in the present context. and to which we first direct attention. The step signal (see Equation 7) takes the electrode in-
(4) K 8.Oldham and J. Spanier, “The Fractional Calculus,” Academic Press, New York, N . Y . . 1974
( 5 ) K. 9.Oldham. J Elecfroanal. Chem. lnterfacial Electrochem., 41, 351 (1973)
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A N A L Y T I C A L C H E M I S T R Y , V O L . 46,
NO.
1 1 , SEPTEMBER 1 9 7 4
stantaneously into a region of complete concentration polarization, so t h a t
c, = C[1
- H(O)]
(17)
This relationship may be combined with Equation 15 to give
for t > 0, whence
(19) Hence the curvature effect enhances the semiintegral a t all times greater than zero, and replaces the horizontal plateau by one of the shape shown as (c) in Figure 5 . For reasons which will be apparent later, it is of interest to follow the surface concentration of Rd during the step experiment we have just described. Combination of Equations 16 and 19 and use of the m, definition gives
as the equation to be solved to give CO’ as a function of time. Standard methods ( 4 ) for the solution of such an “extraordinary differential equation” yield
(21) as the equation giving the surface Rd concentration as a function of time. Note t h a t for r = m, this reduces to CO’ = C V ’ simply. ~- A puise signal is indistinguishable from a step up to time t = t , and Equations 17 through 21 therefore hold up to this instant. Thereafter the potential is switched to the initial potential, a potential so positive that, in the case of reversible reductions only, every molecule of Rd which reaches the electrode interface-’is immediately reoxidized. Hence CO’ is zero after t = t,, and given by Equation 21 before this instant. The expression
incorporates both regimes. It may now be inserted into Equation 16 to provide a complete solution for m in the pulse case. The solution for t < t , is, of course, identical with Equation 19; for t > t , it is the complex expression
li i
t
I
I
0
’P
Figure 5. The shapes predicted for m vs. t curves with step (c) and pulse ( d ) signals if the curvature effect were the sole factor complicating simple behavior Notice that. with the step, the value of rn becomes progressively greater than m,. The medium in which the reductant dissolves determines which of the two pulse curves is actually followed. Compare with Figure 4 and note that for g = -1, the nonfaradaic and curvature effects reinforce each other, but that for g = 1, they are in opposition
sult, for the g = +1 and the g = -1 cases. Notice that, immediately following the pulse time tp, m is predicted to be positive if Rd is soluble in solution, but negative if Rd dissolves in the mercury. These remarks apply, of course, only to reversible electroreductions. Let us now turn to the effect of curvature on m in the case of a ramp signal. The responses to a simple and a capped ramp are identical. We start with Equations 15 and 16 and note that, for a reversible reduction the terms COand CO’are interrelated by the Nernstian expression
In principle, all that needs to be done to determine the m us. t relationship for a ramp signal is to eliminate CO and CO’between Equations 15, 16, and 24. In practice, however, this is beyond the present capabilities of the fractional calculus. We therefore resort to an approximate treatment. Semiintegration of Equation 15 produces
after multiplication by vT>/r. If this result is now subtracted from Equation 15, mr - mq = D [ C - C , , ]- $J’[C - C,,ldt (25) nA F r Thus far our derivation has been exact but, to proceed further, we make use of the approximation
valid if t t, will undoubtedly be negative for reversible electroreductions which yield amalgams, but may be either positive or negative if t h e reduction product dissolves in the electrolytic medium.
Combined with Equation 27, this relationship gives
EXPERIMENTAL A hanging mercury drop electrode, of the conventional design shown as diagram I in Figure 7 , is not surrounded by a diffusion field of truly semiinfinite spherical geometry. This is because of the “shielding effect” of the glass capillary. To overcome this handicap, we have suspended mercury drops from a capillary fit-
1526
* ANALYTICAL CHEMISTRY, VOL. 46, NO.
11, SEPTEMBER 1974
I
a
II
1
I\\
I
The first diagram shows a hanging mercury d r o p of conventional design; spherical geometry does not exist in the neighborhood of the glass/mercury/solution junction. Diagram I I is a cross-section of a nosepiece and shows how it is designed to fit (with the aid of two O-rings) over the end of a conventional glass capillary electrode. The third diagram illustrates the geometry of the electrode Figure 7.
-0 4 I
I
I
0
2
4
I
1
I
6
8
IO
f(secl
Semiintegrated current vs. time curves of 0.1M KNO3 in potential step and pulse Figure 9.
Electrode area: 0.0158 cm2, Step and pulse potentials (mV vs. SCE): ( I ) 200 to 0. ( 1 1 ) -200 to -600
1
0.8p
I
I
I
I
6
8
IO
I
t (sec)
Semiintegrated current vs. time curves of 0.1 M K N 0 3 in simple and capped potential ramps Figure 8.
Electrode area: 0.0158 cm2, Initial and final potentials: 200, 0 mV vs. SCE. Ramp rate: 50 mV/sec. ( I ) Simple potential ramp, ( \ I ) Capped potential ramp
-200
-300
-400
-500
-600
-700
-800
E ( m V vs.SCE)
Figure 10. Neopolarogram and charge vs. potential curve for reduction of TI+ on application of simple potential ramp.
Solution: 5.00mM T I N 0 3 in 0.100M K N 0 3 . Electrode area: 0 0158 cm* Ramp rate: 100 mV/sec. Initial potential: -200 mV vs. SCE. ( I ) Neopolarogram, ( I I ) Charge-potential curve
Table I. Dimensions of Nylon Nosepieces No. 1
No. 2
No. 3
0.0368 30 O
0.0787
0
0.0991 30 O
V(P1)
0.206
30 O 2.01
graphical error in Figure 10 of reference 6. The resistor shown as having a resistance of 830 kilohms actually has a tenfold larger resistance: 8300 kilohms. The author apoligizes for inconvenience caused as a result of this error.)
A (cm ?) 4cm)
0.0158 0.0368
0.0725 0.0787
4.02 0.115
RESULTS AND DISCUSSION
d(cm)
0,0991
ted with a conical “nosepiece” of a design shown in diagram I1 of Figure 7 . The design is such that, when a mercury drop of the correct volume V is exuded, the center of the spherical drop coincides with the apex of the cone and hence the diffusion field of the electrode is accurately spherical and semiinfinite. Diagram 111 of Figure 7 defines the geometric parameters d and B of the nosepiece. The correct volume is related to these parameters by the equation
ad’ v = --[2 + 3 cos0 - COS’B] 24 sin’8 while the exposed area and the drop radius are calculable from the formulas
A=-
2 sin’8
(1
+ cod)
and d r=2 sin8 respectively. Three nylon nosepieces were employed, with dimensions as detailed in Table I. Nosepiece No. 1, with the smallest r value, gives the most pronounced curvature effect, and has therefore been the most heavily utilized in this study. Other experimental conditions were exactly as described in other publications of this series ( I , 2, 6). (Please note a typo( 6 ) K. B. Oldharn, Anal. Chem., 45, 39 (1973)
Nonfaradaic Contribution. The theory relating to the nonfaradaic contribution is best tested with “blank” solutions and Figures 8 and 9 were obtained with such media. With a simple potential ramp, the shape of the m us. t curve for a 0.1M K N 0 3 solution was rather complex as shown in curve I of Figure 8. This complexity arises partly from impurities such as oxygen present in the blank solution and partly from the change of double layer capacitance with potential. For a capped ramp signal, the semiintegrated current of the blank solution decreased after the time t f , see curve I1 of Figure 8, in a fashion similar to the theoretical curve (6) of Figure 3. The step and pulse signal cases gave m us. t curves with shapes as predicted by the theory, as shown in curve I of Figure 9. The curve I1 in Figure 9 is apparently deformed by the presence of traces of oxygen despite prolonged bubbling of argon gas. Electrode Curvature. The electrode curvature effect was tested using neopolarograms and charge us. potential curves from which nonfaradaic and impurity effects had been removed by the subtraction of blanks. Figure 10 indicates a typical neopolarogram and a charge us. potential curve for the reduction of T1+ in 0.1M K N 0 3 on application of simple potential ramp. Curve I shows a pronounced upward slope in the plateau of the neopolarogram. The charge continuously increased with potential as theory predicts.
ANALYTICAL CHEMISTRY, VOL. 46, NO. 1 1 , SEPTEMBER 1974
1527
300
I
60
50
-5 a $
+
(nAFC
40
30 t
0
q/r)(ampere.sec.cm-l x IO')
Relationship between semiintegrated current, charge, and electrode size for reduction of TI+ under complete diffusion control
0
40
Figure 11.
Solution: 5.00mM TIN03 in 0.100M K N 0 3 Ramp rate: 50 rnV/sec. Initial potential: -550 rnV vs. SCE. Electrode area ( c m 2 ) : 0 0.0158, 0 0.0725. A 0.115, Electrode radius ( c m ) : 0 0.0368, 0 0.0787. A 0.0991
80 120 160 q ( p coulomb)
200
Figure 13. Relationship between semiintegrated current and charge under complete diffusion control at application of different initial potentials Solution: the same as in Figure 11. Electrode area: 0.0158 cm'. Ramp rate: 100 mV/sec. Initial potential (mV vs. SCE): 0 -200, A -550.
-a". E
*
30-
E
20 -
0 v
IO -
1
1
I
I
0
2
4
I
I
6
8
3
t (sec)
01
0
I
40
I
80
I
120
I
I60
Figure 14. Semiintegrated current vs. time curves of 5.00mM TI' in 0 . l M K N 0 3 on application of potential step and pulse
I
200
q ( p coulomb) Figure 12. Relationship between semiintegrated current and charge under complete diffusion control at different ramp rates Solution the same as in Figure 11 Electrode area 0 0158 crnL Initial potential -550 mV vs SCE Ramp rate (rnV/sec) 0 50, 0 100, A 200
For simple and capped ramp signals, Equation 27 shows t h a t a plot of m us nAFC + [ q / r ] should yield a straight line whose slope is vD. Alternatively a plot 9f m us. q should produce a straight line whose slope is v D / r and has a n intercept equal to m,. Figure 11 shows a plot of m us nAFC [ q / r ]under complete diffusion control for the reduction of Tlf with different sizes of electrode. The points lie on a straight line independent of the electrode size, though there is some deviation in the case of larger electrodes. This deviation is possibly due to the deformation in shape of larger mercury drops by gravity. The cm sec-l slope of the line in Figure 11 is 4.67 x and compares well with the reference value of 4.27 x cm sec-l calculated from the diffusion coefficient (18.2 x cm2 sec-I) of T1+ in t h e literature (7). Figure 12 shows a plot of m us q for the reduction of T1+ a t different ramp rates. There exists a n excellent linear relationship between m and q under complete diffusion control independent of ramp rates from 50 t o 200 mV sec-l. The
Electrode area: 0.0158 c m Z . Step and pulse potentials: -200 to -600 rnV vs. SCE. ( I ) Potential step, ( I I, I I I ) Potential pulse
30 t
I
+
(7) J Heyrovsky and J Kfita. ' Principles of Polarography Press New York N Y 1965 p 106
1528
Academic
0
I
I
2
4
I
I
6
8
1
10
t (sed
Figure 15.
Semiintegrated current vs. time curves of 5.00mM (COOK)2 on application of potential step and
F e ( l l l ) in 0 . 2 M pulse
Electrode area: 0.0158 c m z . Step and pulse potentials: -100 to -400 mV vs. SCE. ( I ) Potential step, ( 1 1 . I l l ) Potential pulse
slope and intercept of the line are 0.125 sec-1/2 and 37.0 microamplomb, respectively. These values almost agree with the v D / r and m, = nAFCvD values of 0.116
A N A L Y T I C A L C H E M I S T R Y , V O L . 46, NO. 11, SEPTEMBER 1974
60 I
50
1
i
6o so
7
-n 30
.
0
-E'
210
' 0
(E,,,
0' 0
I
I
I 2 t I/? se c1/2)
3
I
12
8
4
-E
16 rnV1/')
20
:4
Figure 18. Relationship between semiintegrated current and potential for reduction of 5.00mM TI+ in 0.1M K N 0 3 in simple potential ramp
Figure 16. Relationship between semiintegrated current and time of TI+ at different concentrations in 0.1M K N 0 3 in potential step Electrode area: 0.0158 cm2. Step potential: -200 to -600 mV vs. Concentration ( m M ) : 0 5.00. A 0.500
Electrode area: 0.0158 cm2. Ramp rate: 100 mV/sec. lnitial potential: -200 mV vs. SCE
70
SCE.
60
-
50
-
40
n
Q
16-
-5
-
5
-
3. 30 E 20
12-
0
3E
-
a-
10
4t 0
0
0
t
I
2
3
t"2(sec1'2
2
I
0
(
Figure 17. Relationship between semiintegrated current and time of 5.00mM Fe(lll) in 0.2M ( C O O K ) 2 in potential step at different potentials Electrode area: 0.0158 cm2. Step potential (mV vs. SCE): 0 -100 to -400, A - 5 0 to -450
and 32.5 microamplomb, respectively, calculated sec by using the literature value of the diffusion coefficient (7). As shown in Figure 13, the relationship between m and q did not depend on the initial potential of the ramp signal. Figures 14 and 2 show m us. t curves for the reduction of T l + in potential step and pulse experiments. Figure 15 shows the same curves for the reduction of Fe(II1). During the potential step, the semiintegral of current continues to enhance with time under conditions in which the concentration of electroactive species is so large t h a t the nonfaradaic effect is negligible, see curves I in Figures 14 and 15. A minimum was observed a t the lower oxidant concentrations as shown in Figure 2. The sign of m following the termination of pulse signals was negative for the reduction of T1+ and positive for that of Fe(III), and the absolute value of each m increased with the pulse duration as shown in curves I1 and 111 of Figures 14 and 15. These results agree well with the pulse theory (Equation 23) because the reduction product of T l + dissolves in the mercury but that of Fe(II1) is soluble in solution. Plots of m us. VT for potential steps were constructed after subtraction of blanks. Figures 16 and 17 are such plots for the reduction of T l + and Fe(III), respectively.
Figure 19. Relationship between semiintegrated current and time of 1.00mM TI+ in 0.1M K N 0 3 without subtracting blank in poten-
tial step Electrode area: 0.115 cm2. Step potential: ca. 0 to -600 mV vs. SCE
There was a linear relationship between m and in each case as predicted by Equation 19. This relationship did not depend on the concentration of electroactive species or on magnitude of the potential step; see Figures 16 and 17. For step and pulse signals, Equation 19 shows that a plot of m us. ~ ' should 7 yield a straight line intersecting the m axis a t m, and with a slope of 2m,yD/rv'ii. The diffusion coefficient of T1+ was calculated to be 21.6 x 10-6 and 17.7 x cm2 sec-l, respectively, from the slope and intersection with the m axis of the upper line in Figure 16. T h a t of Fe(II1) was found to be 7.14 x and 5.40 x respectively, from the line in Figure 17. These values compare well with diffusion coefficient values of 18.2 X cmz sec-' for T1+ and 6.11 X 10-6 cm* sec-1 for Fe(II1) reported in the literature (7, 8) for similar solution conditions. The above experimental results support all predictions concerning the plateau of neopolarograms and m us. t curves. Simple Determination of m,. It is important to be able to determine m, (and hence C) in an actual analysis. For ramp signals, Equation 28 shows that a plot of m us. d m should yield a straight line whose intersection with the m axis is m,. The E l / z potential is first determined by a graphical method ( 2 ) which is inspired from (8) J. J. Lingane, J. Amer. Chem. SOC.,68, 2448 (1946)
A N A L Y T I C A L C H E M I S T R Y , VOL. 46, NO. 11, SEPTEMBER 1974
1529
polarographic practice. This method permits an approximate correction for the nonfaradaic effect by extrapolating the foot of neopolarograms. Figure 18 shows an m us. dE112 - E plot for the reduction of T1+, The points lie on a line which intercepts with the m axis a t 33.5 microamplomb. This m, value compares well with the reference value of 32.5 microamplomb calculated from the literature value of diffusion coefficient ( 7 ) . For a step signal, the theory predicts that the nonfaradaic contribution will be negligibly small after a long enough time. A plot of m us. v‘7 for the curve I1 of T1+ in Figure 2 was constructed without subtracting blank in Figure 19. The points after t = 2.6 sec lie on a straight line whose intersection with the rn axis is 48.5 microamplomb. This value also agrees with the reference value 47.3 microamplomb. It was verified by the above data that these two methods are convenient and satisfactory to estimate rn, value under the conditions of semiinfinite spherical diffusion.
APPENDIX
List of symbols A = electrode area (cm2) C = bulk concentration of Ox (mole cm- 3) CO = the (generally time-dependent) concentration of Ox a t the electrode surface (mole cm- 3 ) CO’= the (generally time-dependent) concentration of Rd a t the electrode surface (mole cm- 3) c = the specific double-layer capacitance (F cm- 2, D = the diffusion coefficient of Ox (cm2 sec- 1) D‘ = the diffusion coefficient of Rd (cm2 sec- 1) d = capillary diameter (cm) = the semidifferentiation operator with respect to dt1/2 time (sec= the semiintegration operator with respect to
time (sec1l2)
dQ’ = the generalized differintegration operator with [d(t - to)lQ respect to t, of order Q, and with a lower limit of t o (sec-Q) E = the (generally time-dependent) electrode potential (V us. some reference) E = the ramp rate (V sec- 1) Ef = a potential in the concentration-polarizing region (V us. same reference)
Eh = the potential defined byEs + m l n nF Eo = initial potential (V)
ble electrode reaction according to simple theory (V)
e- = an electron erfc( ) = the error function complement;
2.J(=
erfc(x) = -
1530
f = the activity coefficient of Ox f’ = the activity coefficient of Rd f ( x ) = an arbitrary function g = a geometric parameter, equal to 1for semiinfinite diffusion in a convex spherical field and to -1 in a concave spherical field (5) 0, t < to H ( ) = the temporal Heaviside function, H(t0) =(1, to
,
i = the (generally time-dependent) faradaic current (A)
I?
In( ) = the natural logarithm function, ln(x) =
-
m = the (generally time-dependent) semiintegral of the faradaic current ( a m p l o m b E A sec1’2) m, = constant defined by nAFCdD (A sec112) h,=, value of rn where this time-dependent function displays a local minimum (A sec1I2) mf = the (generally time-dependent) semiintegral of the nonfaradaic current (A sec112) msp = the (generally time-dependent) enhancement of m due to electrode sphericity (A sec1I2) n = number of faradays to reduce one mole of Ox (equiv mole- I) Ox = a n electroreducible species Q = arbitrary order of differintegration q = the (generally time-dependent) charge (C) qnr = the (generally time-dependent) nonfaradaic charge (C) 9 1 / 2 = charge passed during the 0 < t < t l / 2 interval (C) R = the gas constant (8.31 J K-Imole-l) Rd = reduction product of Ox r = electrode radius (cm) T = absolute temperature (K) t = time after commencement of experiment (sec) to = a n arbitrary time instant (sec) tf = time defined by [Eo - E f ] / E (sec) t,,, = time a t which m = mmln (sec) tp = pulse duration (sec) t l l 2 = time defined by [EO- El/2]/E(sec) tanh( ) = hyperbolic tangent function,
V = pendant volume of mercury (cm3) x = a n arbitrary variable
= ratio of circular circumference to diameter (3.14. . . ) X = an integration variable B = semivertical conical angle of electrode nosepiece
K
(f’a)‘’’
Es = the standard potential of the Ox/Rd couple (V) E1 2 = potential a t which m = lhmc;equal to Eh for reversi-
exp(
F = the faraday constant (0.965 x 105 C/equiv)
ACKNOWLEDGMENT Morten Grenness designed and constructed the three nosepieces and conducted preliminary experiments. His assistance is gratefully acknowledged, as is the financial support of the National Research Council of Canada, and the Defence Research Board.
exp(-X2)dX
) = the exponential function, exp(x) = [2.71828. .
.p
RECEIVED for review September 14, 1973. Accepted February 11, 1974.
A N A L Y T I C A L C H E M I S T R Y , V O L . 46, NO. 11, SEPTEMBER 1974
